The spectral theorem allows for a decomposition of the spectrum of a self-adjoint operator T:D(T)→H, different form the standard one (continuous spectrum σc(T), point spectrum σp(T) and residual spectrum σr(T) absent for self-adjoint operators), consisting in the discrete spectrum
σd(T):={λ∈σ(T)|dim(P(T)(λ−ϵ,λ+ϵ)(H))is finite for some ϵ>0},
plus the essential spectrum
σess(T):=σ(T)∖σd(T).
It is not hard to see that λ∈σd(T) ⇔ λ is an isolated point in σ(T), and as such, λ is an eigenvalue for T with finite-dimensional eigenspace. One has σd(T)⊂σp(T). In general, though, the opposite inclusion fails, for instance because there may be non-isolated points in σp(T).
A third spectral decomposition for T:D(T)→H self-adjoint arises by splitting the Hilbert space into the closed span Hp of the eigenvectors and its orthogonal complement: H=Hp⊕H⊥p. Both Hp∩D(T) and H⊥p∩D(T) are easily T-invariant. With the obvious symbols:
T=T|Hp⊕T|H⊥p.
One calls purely continuous spectrum the set σpc(T):=σ(T|H⊥p), where for simplicity T|H⊥p stands for T|D(T)∩H⊥p here and in the sequel.
Then σ(T)=¯σp(T)∪σpc(T). Note that the latter is not necessarily a disjoint union, and in general σpc(T)≠σc(T).
The fourth spectral decomposition of T:D(T)→H on the Hilbert space H (and even on a normed space), is that into approximate point spectrum
σap(T)
:={λ∈σ(T)|(T−λI)−1:Ran(T−λI)→D(T) does not exist or is not bounded}
and purely residual spectrum
σpr(T):=σ(T)∖σap(T).
The unboundedness of (T−λI)−1 is equivalent to the existence of δ>0 with ||(T−λI)ψ||≥δ||ψ|| for any ψ∈D(T), so we immediately see how the next result comes about, thereby justifying the names: λ∈σap(T) ⇔ there is a unit ψϵ∈D(T) such that
||Tψ−λψ||≤ϵ
for any ϵ>0. For self-adjoint operators the above holds for any λ∈σc(T), but clearly also for λ∈σp(T); since σ(T)=σp(T)∪σc(T) in this case, we conclude σap(T)=σ(T) and
σpr(T)=∅ for every self-adjoint operator.
The last partial spectral classification for self-adjoint operators descends from Lebesgue's theorem on the decomposition of regular Borel measures on R. If T is self-adjoint on the Hilbert space H and μψ is the spectral measure of the vector ψ, we define the sets (all closed spaces):
Hac:={ψ∈H|μψ is absolutely continuous for Lebesgue's measure},
Hsing:={ψ∈H|μψ is singular and continuous for Lebesgue's measure},
Hpa:={ψ∈H|μψ is purely atomic}.
Then we define σac(T):=σ(T|Hac), σsing(T):=σ(T|Hsing) respectively called absolutely continuous spectrum of T and singular spectrum of T.It turns out that σac(T)∪σsing(T)=σpc(T) and¯σp(T)=σ(T|Hpa).